A General Theory for Marine Propellers 



Lq = Lqa + '^^^b = Ae^* , (83c) 



where A = s/h^^ + Lqb and = arctan (Lqb /Lqa)- When these computations are 

 carried out for all the harmonics of the wake field, the resultant blade loading 

 will be the sum of all the loading harmonics. The resultant load distribution 

 over the blade area is the sum of all the load distribution harmonics. 



So far we have ignored the radial fluid velocity component. After the load 

 distribution functions l\p-^(p,0)^ and Ap. (^,^9)^^ have been obtained, Eqs. (72) can 

 be used to obtain the radial induced velocity component. The resultant radial 

 velocity component is the sum of the radial wake component and the propeller- 

 induced radial velocity component. If the resultant radial velocity component at 

 each point P' is large, the mean- line segment shown in Fig. 3 must be taken as 

 the projection of the mean-line segment as traced by a fluid particle through the 

 control point P'. A new value of tan /- at each control point is taken accord- 

 ingly, and the whole computation is repeated. For practical purposes such re- 

 fined computations may not be necessary. 



SOME PRELIMINARY NUMERICAL RESULTS 



Based on the theory and the numerical technique outlined in the previous 

 sections, a computer program is being developed for the problem of designing a 

 propeller and predicting its performance. Some preliminary results will be 

 given for (a) an open-water propeller design with constant chordwise load dis- 

 tribution and (b) the inverse calculation for predicting propeller performance in 

 the steady design condition. 



For the time being the computer program neglects the difference between 

 /Sq and /3 in the integration over the lifting surface (as discussed following Eqs. 

 (73)), and in this case the chordwise integration over the blade is readily carried 

 out functionally. 



Integration in the radial direction is carried out numerically following an 

 integration procedure similar to that described in Ref . 24 for the spanwise inte- 

 gration of a wing. The propeller blade is divided into three regions, as indicated 

 in Fig. 4. Region II extends a short radial distance on each side of the control 

 point P(x,r,0), region I fills the gap between the root section of the blade and 

 region II, and region III extends from region II to the tip of the blade. 



The integrand of Eqs. (19) contains a second- order singularity l/(r-p)^ in 

 region II; hence the division into the three regions is intended to facilitate the 

 evaluation of the finite part of the improper integral in this region. The inte- 

 grands in regions I and III are not singular and can readily be evaluated by nu- 

 merical integration methods. 



Propeller Design Example 



The design example chosen is a propeller with a symmetrical blade outline 

 and constant chordwise load distribution; the specification is similar to that 

 chosen by Pien in Ref. 3 and by Cheng in Ref. 6. 



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